Suppose the market for schools in a small town is of perfect competition. The market demand for school seats, $y$, is given by $y(p)$. The long run average cost function for each school is given by $C(y)$. how do I determine the following?
(i) the long run equilibrium size of each school?
(ii) the tuition price does each school charge
Under perfect competition firms produce in the long run at the point in which average cost is minimum. This is a consequence of the zero profit condition needed in the long run, which in turn determines the number of firms that will operate in the market (because there can't be entry/exit of firms). In your example, this means that each firm will produce a quantity $y^*$ such that $C(y^*)\leq C(y)$ for all $y\geq 0$. Moreover, the zero profit condition also requires that the price in equilibrium to be such that $p=C(y^*)$.
EDIT: Thanks to @jack for pointing out a mistake with the average cost function.